Robust Smoothed Analysis of a Condition Number for Linear Programming

TL;DR

This paper provides a robust smoothed analysis of the GCC-condition number in linear programming, showing that its expected logarithm is bounded by a function of problem size and perturbation radius, extending previous results.

Contribution

It extends smoothed analysis of condition numbers to a broader class of distributions and demonstrates robustness of the bounds under various perturbation models.

Findings

Expected log of condition number is O(mn / σ)

Results apply to radially symmetric distributions with possible singularities

Extends previous Gaussian-based bounds to more general distributions

Abstract

We perform a smoothed analysis of the GCC-condition number C(A) of the linear programming feasibility problem \exists x\in\R^{m+1} Ax < 0. Suppose that \bar{A} is any matrix with rows \bar{a_i} of euclidean norm 1 and, independently for all i, let a_i be a random perturbation of \bar{a_i} following the uniform distribution in the spherical disk in S^m of angular radius \arcsin\sigma and centered at \bar{a_i}. We prove that E(\ln C(A)) = O(mn / \sigma). A similar result was shown for Renegar's condition number and Gaussian perturbations by Dunagan, Spielman, and Teng [arXiv:cs.DS/0302011]. Our result is robust in the sense that it easily extends to radially symmetric probability distributions supported on a spherical disk of radius \arcsin\sigma, whose density may even have a singularity at the center of the perturbation. Our proofs combine ideas from a recent paper of B\"urgisser, Cucker, and Lotz (Math. Comp. 77, No. 263, 2008) with techniques of Dunagan et al.